1. ## Difficult integration

Evaluate $2\pi \int_{\frac{\pi}{8}}^{\frac{3\pi}{8}} \tan x + \sqrt{1 + \sec ^4 x} dx$

i know the final anwser i need to know how to do it. its not for homework or a test i just want to know how to do it

2. lets concentrate on $\int \tan x \sqrt{1 + \sec^4 x}~dx$

Note that this is the same as $\int \frac {\sec^4x \tan x \sqrt{1 + \sec^4 x}}{\sec^4 x}~dx$ ........i multiplied by $\frac {\sec^4 x}{\sec^4 x}$

Now, let $u^2 = 1 + \sec^4 x$

$\Rightarrow 2u~du = 4 \sec^4 x \tan x~dx$

and $\sec^4 x = u^2 - 1$

So our integral becomes:

$\frac 12 \int \frac {u^2}{u^2 - 1}~du$

which is relatively easy to deal with

3. Wow! . . . great soluton, Jhevon!

4. indeed haha! actually, this problem came from http://www.mathhelpforum.com/math-he...plication.html

and look at my long solution, i thought many things at the same time!

5. Hey Jhevon - Ever been call a Mathimagican

6. Originally Posted by Krizalid
indeed haha! actually, this problem came from http://www.mathhelpforum.com/math-he...plication.html

and look at my long solution, i thought many things at the same time!
well, look at that. i never remembered seeing this integral before. how did you?!

it was probably a mistake that i stumbled on it

i just wanted to get rid of the square root. so i made what's underneath it $u^2$. then i realized that i would need a $sec^4 x$ that wasn't there, so i put it there. wonder how it came to me

Originally Posted by danny arrigo
Hey Jhevon - Ever been call a Mathimagican
haha, no, and for good reason

7. Originally Posted by danny arrigo
Hey Jhevon - Ever been call a Mathimagican

I had a student call me that quite often last year.
I'm not sure what he meant by it.

8. Originally Posted by matheagle
I had a student call me that quite often last year.
I'm not sure what he meant by it.
Ya me to, especially with nonlinear differential equations. They keep saying that and "Danny magic."

9. I guess this is magic to them

10. Originally Posted by matheagle
I guess this is magic to them
To paraphrase Arthur C. Clarke: Any sufficiently advanced mathematics is indistinguishable from magic.

And to paraphrase a Moderator: This thread is gettting way off-topic. The question has been answered so ....... thread closed.